Question: All of the 5th grade teachers and students from Almond went on a field trip to an archaeology museum. Tickets were $$5.00$ each for teachers and $$2.50$ each for students, and the group paid $$35.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$10.00$ each for teachers and $$12.50$ each for students, and the group paid $$115.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5x+2.5y = 35}$ ${10x+12.5y = 115}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-10x-5y = -70}$ ${10x+12.5y = 115}$ Add the top and bottom equations together. $ 7.5y = 45 $ $ y = \dfrac{45}{7.5}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {5x+2.5y = 35}$ to find $x$ ${5x + 2.5}{(6)}{= 35}$ $5x+15 = 35$ $5x = 20$ $x = \dfrac{20}{5}$ ${x = 4}$ You can also plug ${y = 6}$ into $ {10x+12.5y = 115}$ and get the same answer for $x$ ${10x + 12.5}{(6)}{= 115}$ ${x = 4}$ There were $4$ teachers and $6$ students on the field trips.